Mixed mode fracture in power law hardening materials for plane stress. (June 2020) - Loghin, Adrian
Abstract: The classic nonlinear fracture problem of a fully yielded, mixed mode stationary crack in a power law hardening material for conditions of plane stress under small-scale yielding is reconsidered. It has been determined that two different asymptotic solutions are required to represent the full range of mixed mode loading. Neither asymptotic solution has the double root of the linear elastic counterpart, i.e., the nonlinear plane stress problem does not have a mixed mode asymptotic solution. The mode II dominant asymptotic solution consists of two terms. The leading term is the pure mode II HRR term, while the second term is symmetric with an eigenvalue slightly weaker than the HRR eigenvalue. This two-term solution applies to a relatively large range of mixed mode loading. The mode I dominant asymptotic solution also consists of a symmetric and an antisymmetric term with different eigenvalues, and has a limited range of applicability near mode I. The pure mode I HRR term is the symmetric term. Contrary to expected behavior based on energy considerations and experience with higher order solutions, the antisymmetric term has an eigenvalue that is stronger than the HRR eigenvalue. This antisymmetric asymptotic solution, which cannot exist without the presence of the mode I HRR term, depends on two small parameters: the distance from the crack tip, r, and the ratio of mode II to mode I loading, K 2 / K 1 . The interpretation is that this two-term asymptotic solution exists for small r in the limit as K 2 / K 1 approaches zero. An unusual feature of the second term is that it does not exist in the limit as r approaches zero, and therefore from a mathematical point of view this term does not cause the J-integral to be infinite. The asymptotic results are confirmed with full-field finite element analysis by using the J2 deformation theory of plasticity using a computational domain that covers eleven decades of radial detail. This validates the asymptotic solutions and shows that a two-parameter fracture theory can be used very near mode I and near mode II. The transition from one asymptotic solution to the other, which is demonstrated to occur near mode I, gives rise to a loss of dominance of these two-term asymptotic solutions. The hardening exponent, "n", plays an important role in the ranges of validity of the two asymptotic solutions. Finally, the asymptotic solutions are shown to agree with solutions from the non-hardening limit, and the comparisons are consistent with those of the full-field results.